Solution Methods And Complex Responses For Nonlinear Aeroelastic Systems

Almost all dynamical systems encountered in practical engineering are inherently nonlin-ear. Linear systems are some simple approximations of the corresponding nonlinear systems.Essentially, many characteristics of the nonlinear systems cannot be described by the linearones. However, due to the existence of the nonlinear term that nullifies the additivity andhomogeneity properties pertaining to the linear systems, exact analytical solutions are rarelyavailable. Therefore, researchers have to resort to approximation methods of solution of non-linear dynamical systems.This thesis is devoted to the study of solution methods and complex dynamical behaviorsof a typical two dimensional airfoil system undergoing subsonic flow. In the aspect of compu-tational methods, the simple time domain collocation (TDC) method is proposed. The relation-ship between the TDC method and the famous high dimensional harmonic balance (HDHB)method is studied. The HDHB method is demonstrated to be essentially a collocation methodin disguise, instead of a variant of the harmonic balance method. Based on this awareness, thealiasing phenomenon of the HDHB method is clearly explained. In the course of investigatingthe two dimensional airfoil, various methods, namely, the time domain collocation method,the harmonic balance method, the high dimensional harmonic balance method as well as thefourth-order Runge-Kutta (RK4) method are used to analyze the bifurcation phenomenon, thelimit cycle oscillation, the quasi-periodic motion, chaos, and chaotic transients. Concretely, thefollowing contents are studied.Using the Duffing oscillator as the prototypical system, we propose the time domaincollocation (TDC) method. The equivalence of the famous high dimensional harmonic balance(HDHB) method and the TDC method is rigorously demonstrated. Based on the awareness thatthe HDHB method is a collocation method instead of a variant harmonic balance method, wedevelop an extended TDC method through collocating at more points than unknowns. The ex-tended TDC method can effectively remove the nonphysical solutions arising out of the HDHBmethod.The periodic solutions of a two dimensional airfoil subjected to subsonic flow are ob-tained using the TDC method. When the HDHB method is employed to solve nonlinear dy-namical systems, there will be additional solutions which are physically meaningless. It isfound that the occurrence of the nonphysical solutions is due to the fact that the high orderharmonics are aliased to low order harmonics in some undiscovered way. This is referred toas aliasing phenomenon. In this study, the mechanism of the aliasing phenomenon has been revealed, and the aliasing rules have been discovered.A fast harmonic balance technique has been proposed. Conventionally, the harmon-ic balance method first transforms a nonlinear dynamical system into a system of nonlinearalgebraic equations (NAEs); consequently, the NAEs are solved by Newton-Raphson (NR)method. In this study, we have derived the explicit Jacobian matrix of the system to acceleratethe computational efficiency of the HB method. Moreover, the spectral analysis is applied tothe numerical solutions to provide insight into the distribution of the dominant frequencies, soas to provide a reasonable suggestion for the truncation of the Fourier series expansion in theHB method.Chaotic motions of a two dimensional airfoil with cubic and freeplay nonlinearities inthe pitch degree of freedom are investigated via numerical integration method. For the air-foil with cubic nonlinearity, the governing ODEs are solved by RK4directly. For the freeplaynonlinearity case, the RK4cannot capture the aeroelastic response accurately due to its inca-pability of detecting the switching points of the freeplay. To resolve this problem, the RK4method is used with the aid of the Henon’s method (referred to as the RK4Henon method) toprecisely predict the freeplay’s switching points. The comparison of the classical RK4and theRK4Henon methods is carried out in the analyses of periodic motions, chaos, and long-livedchaotic transients. For both cases, complex dynamical behaviors are revealed and identifiedthrough the means of bifurcation diagrams, the phase portraits, the amplitude spectra and thePoincare maps. An interesting long-lived chaotic transient is observed in the two dimension-al airfoil system for the first time. In addition, the effects of various system parameters areinvestigated.